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F-invariant submanifolds of Kaehlerian product manifold

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In this paper, the geometry of F-invariant submanifolds of a Kaehlerian product manifold is studied. The fundamental properties of these submanifolds are investigated such as pseudo umbilical, curvature invariant, totally geodesic, mixed geodesic submanifold and locally decomposable Riemannian product manifold. | Turk J Math 28 (2004) , 367 – 381. ¨ ITAK ˙ c TUB F-Invariant Submanifolds of Kaehlerian Product Manifold Mehmet At¸ceken Abstract In this paper, the geometry of F-invariant submanifolds of a Kaehlerian product manifold is studied. The fundamental properties of these submanifolds are investigated such as pseudo umbilical, curvature invariant, totally geodesic, mixed geodesic submanifold and locally decomposable Riemannian product manifold. Key Words: Kaehlerian Product Manifold, Mixed Geodesic Submanifold, Locally Decomposable Riemannian Manifold and Constant Holomorphic Sectional Curved Manifold. 1. Introduction The geometry of submanifolds of a Kaehlerian product manifold is an interesting subject which was studied by many geometers. Partially, the geometry of CR-submanifolds of a Kaehlerian product manifold was studied by M. H. Shahid [8] and he had many interesting results of this submanifold. Also, the geometry of CR-submanifold of any Kaehlerian manifold was studied by Bejancu A., [2] and Chen B. Y. [3, 4]. The object of this note is to study F -invariant submanifolds of a Kaehlerian product manifold. In this paper, we have researched the fundamental properties of F-invariant submanifolds of a Kaehlerian product manifold. We think interesting results such as Theorem 4.5, Theorem 4.6 and Theorem 4.7 are obtained in this paper. We show that Mathematics Subject Classification (2000): 53C42, 53C15. 367 ATC ¸ EKEN a F-invariant submanifold of a Kaehlerian product manifold and their distributions have the same properties. 2. Preliminaries ¯ be a m-dimensional manifold Let M be an n-dimensional Riemannian manifold and M ¯ becomes a Riemannian submanifold of M with isometrically immersed in M . Then M ¯ ⊥ the Riemannian metric induced by the Riemannian metric on M . We denote by T M ¯ and by g both metrics on M and M ¯ . Also, we denote by ∇ ¯ and ∇ normal bundle to M ¯ and M , respectively. Then the Gauss formula is given the Levi-Civita connections on